Analog of the Birkhoff’s ergodic theorem for the sequence of squares

No – the sequence of squares is universally bad which was proved by Buczolich and Mauldin. I will quote from Tom Ward’s review of their paper Divergent square averages, Ann. of Math. (2) 171 (2010), no. 3, 1479–1530.

A consequence of J. Bourgain’s work [Inst. Hautes Études Sci. Publ. Math. No. 69 (1989), 5–45; MR1019960] is an ergodic theorem along squares, answering earlier questions of Bellow and Furstenberg: If $(X,mathcal B,T,mu)$ is a measure-preserving system, then the non-conventional ergodic averages
$$
frac1{N} sum_{n=0}^{N-1} f(T^{n^2} x)
$$

converge almost everywhere for $fin L^p$ with $p>1$. Here a comprehensive – and negative – answer is given to his question of whether the result extends to all of $L^1$. The authors show that the sequence $(n^2)$ is universally bad: for any ergodic measure-preserving system there is a function $fin L^1$ for which the above averages fail to converge as $Ntoinfty$ for $x$ in a set of positive measure.

PS The Birkhoff theorem does not apply to your “particular case” as it requires the presence of a finite invariant measure.

If $X=mathbb{Z}$, $mu$ is the counting measure, and $T$ is the shift operator given by $Tf(x)=f(x+1)$, then for all real $pge1$, $finell^p(mathbb{Z})$, and $xinmathbb{Z}$, by Hölder’s inequality,
$$
|mathcal{A}_N f(x)|le frac1N,sum_{n=0}^N|f(x+n^2)|
lefrac1N,|f|_p,(N+1)^{1-1/p}to0
$$

and hence $mathcal{A}_N f(x)to0$ as $Ntoinfty$.

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